Data Sufficiency – Problems That Need Pre-Work 2

In the previous Quantitative post , we saw how seemingly tough and time-consuming Data Sufficiency problems usually require a certain amount of pre-work.

In most cases if the pre-work is done properly, you will precisely know what information is required to answer the question even before you go to the statements. The GMAT question below is another one of such problems.

Each data sufficiency problem below consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you are to option

A if statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
B if statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
C if BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient;
D if EACH statement ALONE is sufficient to answer the question asked;
E if statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Positive integer k has exactly 2 positive prime factors, 3 and 7. If k has total of 6 positive factors, including 1 and k, what is the value of k?

(1) 3² is a factor of k
(2) 7² not a factor of k

As soon as you read such a question — one that gives so much specific information — you need to realize that this is a type that needs pre-work.

Every number can be written in terms of the powers of its prime factors. For example, 36 is 2²*3² and 90 is 2*3²*5.

So if a number when written in terms of the powers of it’s prime factors is represented as a^m. b^n, then the number of factors it has is equal to (m+1)(n+1).

You just add one to every power and multiply all of them. For example 90=2¹*3²*5¹ will have (1+1)(2+1)(1+1) or 12 factors.

In this case the number k can be written as 3^m*7^n

Since the number of factors of k is given to be equal to 6, (m+1)(n+1) = 6 = 2*3 = 3*2.

So either (m+1)(n+1) = 2*3, in which case m=1 and n=2 and hence, k = 3*7² — (I),

So even, before we reach the statements we know that there are only two possibilities for the value of k.

Statement (1) says 3² is a factor of k, so k has to be (II).

Statement (2) says 7² is not a factor of k, so K cannot be (I), it has to be (II).

So, each statement alone gives you the answer, so option (D) is the one to choose.

As you can see this is a very methodical way of unlocking the problem, like precisely narrowing down the options to a number lock, as opposed to straightaway jumping to options to use the statements to randomly try out possibilities.

When you do that you are never really sure whether you have answered the question correctly. It usually takes longer, making (C), both statements are needed, a very attractive option.

Also, please do not limit this approach to Data Sufficiency.

Even on Problem Solving questions, do not start writing the moment you start reading the question — the moment test-takers start reading a question, they simultaneously start,

duplicating information given on the screen on to their scratch pads,

take the first unknown thing they encounter as x,

start forming equations with every sentence they read.

A better way is to

wait till you reach the end of the problem,

put all the information together,

if you want to solve it using equations, then take what is required as x, and write everything else in terms of that

This is one of those rare GMAT Quantitative problems that requires you to remember a formula, thankfully such questions are far and few in between! Let me know, via the comments, if you want to learn how the formula used in this problem came about in the first place.

Also, do not hesitate to post queries in the comments section, in case you need any elaboration or clarification.

For example in this case the problem mentions the told number of factors, so you should pause there and ask yourself what is the concept/formula associated with total number of factors.

When you start evaluating information given in the question in this manner your accuracy levels will go up.

Anyway, this is one of those rare GMAT problem (Geometry ones apart) that needs a formula.

In case you do not know this formula, you can still do this conceptually.

3 & 7 are the prime factors and there are 6 prime factors. So this number is essentially 3^m * 7^n.

So what can the factors possibly be using statement 1?
Since 3^2 is a factor and other existing factors are 1, 3 and 7, 3.7 and 3^2 *7 will also be factors, making it 6 factors in total. If 7 has a power greater than 1, then the number of factors will be greater than 6. So number has to be 3^2*7.